Critical curve of two-matrix models $ABBA$, $A\{B,A\}B$ and $ABAB$, Part I: Monte Carlo

Abstract

For a family of two-matrix models \[ \frac{1}{2} \operatorname{Tr}(A^2+B^2) - \frac{g}{4} \operatorname{Tr}(A^4+B^4) - \begin{cases} \frac{h}{2} \operatorname{Tr}( A BA B) \\ \frac{h}{4} \operatorname{Tr}( A BA B+ ABBA ) \\ \frac{h}{2} \operatorname{Tr}( A B BA ) \end{cases} \] with hermitian $A$ and $B$, we provide, in each case, a Monte Carlo estimate of the boundary of the maximal convergence domain in the $(h,g)$-plane. The results are discussed comparing with exact solutions (in agreement with the only analytically solved case) and phase diagrams obtained by means of the functional renormalization group.